The Remainder and Factor Theorems The Remainder Theorem
- Slides: 15
The Remainder and Factor Theorems
The Remainder Theorem If a polynomial f(x) is divided by (x – a), the remainder is the constant f(a), and f(x) = q(x) ∙ (x – a) + f(a) where q(x) is a polynomial with degree one less than the degree of f(x). Dividend equals quotient times divisor plus remainder.
The Remainder Theorem Find f(3) for the following polynomial function. f(x) = 5 x 2 – 4 x + 3 f(3) = 5(3)2 – 4(3) + 3 f(3) = 5 ∙ 9 – 12 + 3 f(3) = 45 – 12 + 3 f(3) = 36
The Remainder Theorem Now divide the same polynomial by (x – 3). 5 x 2 – 4 x + 3 3 5 – 4 3 15 33 5 11 36
The Remainder Theorem f(x) = 5 x 2 – 4 x + 3 f(3) = 5(3)2 – 4(3) + 3 f(3) = 5 ∙ 9 – 12 + 3 f(3) = 45 – 12 + 3 5 x 2 – 4 x + 3 3 5 – 4 15 5 11 3 33 36 f(3) = 36 Notice that the value obtained when evaluating the function at f(3) and the value of the remainder when dividing the polynomial by x – 3 are the same. Dividend equals quotient times divisor plus remainder. 5 x 2 – 4 x + 3 = (5 x + 11) ∙ (x – 3) + 36
The Remainder Theorem Use synthetic substitution to find g(4) for the following function. f(x) = 5 x 4 – 13 x 3 – 14 x 2 – 47 x + 1 4 5 5 – 13 – 14 20 28 7 14 – 47 56 9 1 36 37
The Remainder Theorem Synthetic Substitution – using synthetic division to evaluate a function This is especially helpful for polynomials with degree greater than 2.
The Remainder Theorem Use synthetic substitution to find g(– 2) for the following function. f(x) = 5 x 4 – 13 x 3 – 14 x 2 – 47 x + 1 – 2 5 5 – 13 – 14 – 47 1 – 10 46 – 64 222 – 23 32 – 111 223
The Remainder Theorem Use synthetic substitution to find c(4) for the following function. c(x) = 2 x 4 – 4 x 3 – 7 x 2 – 13 x – 10 4 2 – 4 8 2 4 – 7 – 13 – 10 16 36 92 9 23 82
The Factor Theorem The binomial (x – a) is a factor of the polynomial f(x) if and only if f(a) = 0.
The Factor Theorem When a polynomial is divided by one of its binomial factors, the quotient is called a depressed polynomial. If the remainder (last number in a depressed polynomial) is zero, that means f(#) = 0. This also means that the divisor resulting in a remainder of zero is a factor of the polynomial.
The Factor Theorem x 3 + 4 x 2 – 15 x – 18 x– 3 3 1 1 4 3 7 – 15 – 18 21 18 6 0 Since the remainder is zero, (x – 3) is a factor of x 3 + 4 x 2 – 15 x – 18. This also allows us to find the remaining factors of the polynomial by factoring the depressed polynomial.
The Factor Theorem x 3 + 4 x 2 – 15 x – 18 x– 3 3 1 1 4 3 7 – 15 – 18 21 18 6 0 x 2 + 7 x + 6 (x + 6)(x + 1) The factors of x 3 + 4 x 2 – 15 x – 18 are (x – 3)(x + 6)(x + 1).
The Factor Theorem (x – 3)(x + 6)(x + 1). Compare the factors of the polynomials to the zeros as seen on the graph of x 3 + 4 x 2 – 15 x – 18.
The Factor Theorem Given a polynomial and one of its factors, find the remaining factors of the polynomial. Some factors may not be binomials. 1. x 3 – 11 x 2 + 14 x + 80 x– 8 2. 2 x 3 + 7 x 2 – 33 x – 18 x+6 (x – 8)(x – 5)(x + 2) (x + 6)(2 x + 1)(x – 3)
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